An approximate general formula to calculate the directivity of an array of antennas by a cylindrical near field acquisition is proposed. We demonstrate the role of and scan area truncation effects. The derivation is based on the theory of cylindrical wave expansion of electromagnetic fields

Besada Sanmartin, Jose Luis

Castelo Boso, Alcino

Haro Ariet, Leandro de

Besada Sanmartín, José Luis

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Aperture Field Reconstruction by Calculated Cylindrical Near Field

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APERTURE FIELD RECONSTRUCTION 
BY CALCULATED CYLINDRICAL NEAR 
FIELD 
Alcino Castelo Boso, Leandro de Haro Ariet, and 
Jose Luis Besada 
Departamento Sistemas, Senates y Radiocomunicaciones, ETSI 
Telecomunicacion, Universidad Politecnica de Madrid, Ciudad 
Universitaria, Madrid 28040, Spain; Corresponding author: 
ABSTRACT: A method for determining the aperture field distribution 
of an array antenna from near-field measurement over the surface of a 
light circular cylinder enclosing the antenna is applied here. The ap-
proach relies on the concept of information content of the field. The 
truncation problem is solved by picking up the information that is lost 
due to the finite size of scanning area, in points of the space reachable 
by the measurement system. In this article, we rebuild the field at the 
aperture from near-field cylindrical calculated on the surface of mea-
sure, with dependence on the variable phi, and without dependence on 
the variable phi. 
Key words: antenna measurement; aperture field; variable phi 
1. INTRODUCTION 
The method for determining the aperture field distribution of an 
antenna from near-field measurements over the surface of a right 
circular cylinder enclosing the antenna has been widely employed 
to determine the far-field patterns of antennas under test. This is 
derived from the wave modal expansion theory that has already 
been employed in near-field to far-field transformation techniques 
based on the three-dimensional vector cylindrical wave expansion 
of an electromagnetic field. 
The truncation error is due to the finite size of the scanning 
surface, for the minimization of the error source, in the develop-
ment of near-field measurements. The standard approach to reduce 
the truncation error is to extend the measurement surface up to 
include the entire region wherein the field radiated by the antenna 
under test is significant The a priori information consists in 
the overall dimension of the antenna, in its position in the space 
and in further information on the shape of the antenna. 
It is possible to recover the information associated with the 
samples falling in the part of the scanning surface external to the 
actual scanning area by measuring the field at other points around 
the AUT that area reachable by the scanning set-up. The samples 
of the representation falling inside the actual scanning area are 
directly measured. To recover the external samples, measurements 
having the same information content of the lost samples are 
performed in an area reachable by the measurement set-up, usually 
in a frame around the available scanning area. Then, a linear 
system relating the samples on the ideal scanning surface and the 
data measured on the frame is implemented by means of aperture 
field reconstructions near-field transformation 
In this article, we reconstruct the field at the aperture from 
near-field cylinder calculated on the surface of measure, with 
dependence on the variable phi, and without dependence on the 
variable phi. 
2. FORMULATION 
In a linear, isotropic region containing no sources, a solution \j/n to 
the scalar wave equation in the cylindrical coordinate system is 
i//;,(r,4>,z) = z;,(Ar)exp(jn<#>)exp( -jhz) (1) 
where n is any integer, h is any real number, and A is 
k r J^J? h<k 
The vectors Mn and N„ are written as 
/ jn . „3Z'\ 
M'nh{r^,z) = y r-Z'n{Ar) - 4>—j exp(j'«<«exp( - jhz) (3) 
/ jh flZJ, jih A2 . \ 
iV<„0-Az) = { - f k ^ r + ^z'n + zyZ^Arj jexpOn^exp 
(-M) (4) 
The function ZJ,(Ar) is any one of the four cylindrical Bessel 
functions given by 
Zl(Ar) = J„(Ar),Zl(Ar) = Yn(Ar),Zl(Ar) = H[y>(Ar) ,Zt(Ar) 
= fl®(Ar) (5) 
In the present case, a solution for E is required, which is valid in 
the region external to a small cylinder containing all sources 
defined by r > ra. In order for is-field to satisfy the radiation 
condition at infinity, the large argument asymptotic expansion of 
the cylindrical Bessel function must represent surfaces of constant 
phase which propagates in the positive radial direction. For time 
variations of the form exp(jW), the only one of these functions 
which satisfies this condition is a Hankel function of the second 
kindfl^'fAr) [2]. The large argument asymptotic expansion of this 
function is 
HfKAr) ^ / ' + U ^ r j e x p ( - JAr); /•->«, 
The general linear combination of the vectors M and N will involve 
an integral over all real h and a sum over all integer n. Thus, in this 
case, the general solution for E-field can be written as 
E{r,cb,z)= E {aMMth{r,<p,z) + bn(h)Nth(r^,z))dh 
(6) 
where ajji) and bn(h) are the complex amplitude weighting func-
tions of the vectors M and N, respectively [2]. The corresponding 
solution for H follows from Maxwell's equation 
H = (— l/jwfJu)V X E and is given by 
ff(r,cM = T ^ r S (bn(h)Mth(r,<p,z) + an{h)NAnh{r,^z))dh 
(7) 
For magnetic and electric components can be written as 
#z(r,(kz) = ^ - X I A2a„(h)Hn(2\Ar)ei"^Jbdh (8) 
*•(>•.*.*)= ^ 2 I (*.w* 
BHn(2)(dr) nh ,,. . . , ., 
X ^ ^ - an{h)—HrP{Ar) \^-J,Kdh (9) 
Hr(r,<j},z) =-. 2 
JVC/A _ 
X - &„(/*)£ — + a„(h)(jhk) A ^ \e>n*-1,adh 
dr 
(10) 
radiated by the dipole or array of dipoles on any cylinder can be 
expressed in terms of b0(h) as 
£ Z ( / I A . Z L 
2TTkH\f\Ar) 
M2,(Ar0) Ez(r0,<t>,z)dzdh (18) 
and 
_ *>nzkuhH<$\kr) 
£
*
(
'"
1
'^
1
'
Zl)
 ~ A2ff<2)(Ar0)fa-0 Ez(r0,<f>,z)dzdh (19) 
i/ is an integer different of zero. 
Whenever, if the z component of £-field radiated by the center-
fed dipole of length L with sinusoidal current distribution is 
E, = -
•JloV/e-*" —fat P ~i*« 
4TT \ Sx 
(kL\e->kR\ 
Ez(r0,<t>,z) = E | bn(.h)jH®(Arde»'+-*"dh (11) 
E4,{r0,<f>,z) = X 
nA
 m dH^(Ar0)' 
b„(h)wH*>(ArJ - aM- dr 
X e>'"t'~Jladh (12) 
* = V^T z ; «i / - + U - 2 ; * = rz + z + 
(21) 
where fc = 2-7T/A is the wavenumber, 17 is intrinsic impedance of 
free space and I0 is maximum current, the z component of ii-field 
radiated [Eq. (20)] by the center-fed dipole and array of dipoles 
linearly along the axis z, is independent of </>. 
and their inverse 
a„(h) •jap &H®{®r) Hlr,^,z)e'
jn
^^
hzd^dz (13) 
$2ff«(2)($r0) 
b
nih) = ^,2rr„(2)/^„T [ | Ez(r0,(f>,z)exp( - jn4>)exp(jhz)d(j)dz 
(14) 
3. DISCUSSION AND NUMERICAL EXAMPLES 
In this section, we report some numerical examples assessing the 
effectiveness of the work; The AUT consists of a linear array of 
3 X 25 dipoles oriented along the z axis, with equal excitation, and 
uniformly spaced on a square lying in the xz plane and A/2 
equispaced. Each dipole is fed by a unit electric current. The 
near-field component along £ of the electric field is measured by a 
probe on a scanning surface placed at a distance d = 54A from the 
AUT. The reconstruction has been made considering 153 points 
along 4> and 145 points along the z axis on the surface of a 
nh
 m 3#,(,2)(Ar0) 
K(h)^?(Ar0) ~ a„(h)- dr 
1 
4TT: 
E^r^,z)e-^^hzd^dz (15) 
Let 
exp(-jn<j))d(j) = \ Q
 n # 0 (16) 
then, bn becomes b0(b„ = 0 for n ¥= 0) only: 
2-rrfc 
b„(h) = b0(h) = A2Ho(2)rAr \ Ez(r0,<f>,z)exp(jhz)dz (17) 
Since the z component of the magnetic field, Hz(r0,4>,z), is zero in 
this cases, it follows from, (13) that the other amplitude weighting 
function a0(h) is correspondingly zero. Thus the normalized fields 
Figure 1 Reconstructed and calculated field (ET) with dependence of the 
component phi. [Color figure can be viewed in the online issue, which is 
available at www.interscience.wiley.com] 
60 80 100 120 140 160 
Theta in degree 
Figure 2 Calculated and reconstructed field (Ez) of array dipoles without 
dependence of the component phi. [Color figure can be viewed in the 
online issue, which is available at www.interscience.wiley.com] 
cylinder, and the practical upper bounds on the sample intervals 
are A.(j> = — and Az = -rj-, and a dipole and array of dipoles 
of length L with sinusoidal current distribution, linearly along the 
axis z. 
Then we will see diagrams of radiation in the near-field com-
ponent z, calculated and reconstructed, for an array of dipoles and 
for a dipole, with dependence and independence of the variable 
phi. 
We see in Figure 1 diagram of the near-field calculated and 
reconstructed for an an'ay of dipoles distributed rectangular taking 
into account the electric field component phi. While in Figure 2 we 
see the near-field diagram calculated and reconstructed for an array 
of dipoles distributed linearly therefore not taking into account the 
electric field component phi. 
We see in Figure 3 diagram of the near-field calculated and 
0 
10 
20 
30 
1
 ^%^ 
rj_0_ Reconstructed 
J j _ Calculated 
i i 
S S ^ ^ - M E ! ^ ^ 
1 
1 
\ ~ 
V 
1 
0 
20 
40 
60 
-
tP* 
1 |f 
y 
4/ 
1 1 
I ' 
\ 
1 
-
-
-
v; -
z in wavelengths 
- - - • reconstructed 
— calculated 
Figure 4 Calculated and reconstructed field (E.J of dipole without de-
pendence of the component phi. [Color figure can be viewed in the online 
issue, which is available at www.interscience.wiley.com] 
reconstructed for a dipole distributed rectangular taking into ac-
count the electric field component phi. While in Figure 4 we see 
the near-field diagram calculated and reconstructed for a dipole 
distributed linearly therefore not taking into account the electric 
field component phi. 
4. CONCLUSION 
We have good results in the determination of the aperture field 
distribution of an array antenna from near-field measurement over 
the surface of a right circular cylinder enclosing the antenna. 
The method takes advantage of the possibility, present in most 
of the scanning set-ups, to move the probe not only on the scanning 
surface, but also along the axis perpendicular to it. This possibility 
allows us to modify the standard cylindrical measurement proce-
dure in order to recover the information contained in the part of the 
scanning surface external to the actual scanning area. 
REFERENCES 
O.M. Bucci and C. Gennarelli, Use of sampling expansions in near-
field-far-field transformations: The cylindrical case, IEEE Trans Anten-
nas Propagat 36 (1988), 830-835. 
W.M. Leach, Jr and D. Paris, Probe compensated near-field measure-
ments on a cylinder, IEEE Trans Antennas Propagat AP-21 (1973), 
435-445. 
O.M. Bucci, A new method for avoiding the truncation error in near-
field antennas measurements, IEEE Trans Antennas Propagat 54 (2006), 
2940-2951. 
A. Yaghjian, An overview of near-field antenna measurements, IEEE 
Trans Antennas Propagat AP-34 (1986), 30-45 
Figure 3 Reconstructed and calculated field (£z) with dependence of the 
component phi. [Color figure can be viewed in the online issue, which is 
available at www.interscience.wiley.com] 


Aperture Field Reconstruction by Calculated Cylindrical Near Field

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